testing for unit roots in seasonal time series

Testing for Unit Roots in Seasonal Time SeriesAuthor(s): D. A. Dickey, D. P. Hasza, W. A. FullerSource: Journal of the American Statistical Association, Vol. 79, No. 386 (Jun., 1984), pp. 355-367Published by: American Statistical AssociationStable URL: http://www.jstor.org/stable/2288276Accessed: 12/05/2010 02:03

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Testing for Unit Roots in Seasonal Time Series

D. A. DICKEY, D. P. HASZA, and W. A. FULLER*

Regression estimators of coefficients in seasonal autoregressive models are described. The percentiles of the distributions for time series that have unit roots at the seasonal lag are computed by Monte Carlo integration for finite samples and by analytic techniques and Monte Carlo integration for the limit case. The tabled distributions may be used to test the hypothesis that a time series has a seasonal unit root.

KEY WORDS: Time series; Seasonal; Nonstationary; Unit root.

1. INTRODUCTION

Let the time series Y, satisfy

yt = ady,-d + et, t 1, 2, . . ., (1.1)

where Y-d+ , Y-d+ 2,. . . , Yo are initial conditions and the e, are iid (0, c2) random variables. Model (1.1) is a simple seasonal time series model in which monthly data are represented by d = 12, quarterly data by d = 4, and so on.

We consider several regression-type estimators of ad

and compute percentiles of their distributions under the hypothesis that ad = 1. This hypothesis states that the seasonal difference Y, = Yt - Y- d is white noise for model (1.1). Extensions of model (1.1) permit a variety of applications.

2. ESTIMATORS

Our first estimator of Oad iS the least squares estimator defined as

n -1 n

ad = Y,-d2 Yt-dYt. (2.1) t=1 t=1

If the initial conditions are fixed and et is normal, Ad iS

the maximum likelihood estimator. The Studentized regression statistic for testing the hypothesis Ho: ad =

1 is

Td ( = Yd) J (aLd - 1), (2.2)

where n

S2 = (n - 1)' Y (Yt - adY,-d) (2.3) t= 1

* D. A. Dickey is Associate Professor, Department of Statistics, North Carolina State University, Raleigh, NC 27695-8203. D. P. Hasza is a statistician at Boeing Computer Services, P.O. Box 7730, Wichita, KS 67277-7730. W. A. Fuller is Professor, Department of Statistics, Iowa State University, Ames, IA 50011. The authors are indebted to an anonymous referee for helpful suggestions.

The statistics ?ld - 1 and 'd are standard output from a computer regression of Y. = Y= - Y, -d on Y_d. Dickey (1976), Fuller (1976), and Dickey and Fuller (1979) discussed a1 and Ti; M.M. Rao (1978) and Evans and Savin (1981) discussed a,.

An alternative model for seasonal data is the stationary model in which the observations satisfy

yt = OtdYt-d + e,, I ad I < . (2.4)

Furthermore, for normal stationary Yt satisfying (2.4), Y, - ad Yt +d is a NID(O, a 2) time series. That is,

Yt = tdyt+d + Vt, Vt - NID(O, cr 2). (2.5)

Motivated by (2.4) and (2.5), we suggest an alternative estimator of td, which we call the symmetric estimator. The symmetric least squares estimator, &d, pools information from the regression of Y, on Yt+d and the regression of Y, on Y,d. Let

d= (2 Ytytd) /(4 ( Y,2 + Y-d ))2, (2.6)

and define the associated Studentized statistic as

d= 2 [{E (Yt2 + Y,-d 2) S2 (&d 1), (2.7)

where n

S2 = (2n - 1)' E [(Y, - dY,-d)2 t= I

+ (yt-d -adYt)]. (2.8)

It follows from the definitions that - 1 I &d < 1 and that

2 itd = -[(2n - 1)(1 - &Ad)]i(1 + &d)<1,

where rd is a monotone function of &d. Thus tests based on Td will be equivalent to tests based on &d.

The statistics &d and 2 - id can be obtained using standard regression programs. Table 1 gives two columns ap- propriate for the regression computation when d = 2 and n = 5. A * indicates a missing value. Thus an observation with a missing value for either the independent or the dependent variable is not used in the regression. The In- dependent Variable column is the Dependent Variable column lagged by the seasonal d. The ordinary regression of the first column on the second column gives the estimator a2

? Journal of the American Statistical Association June 1984, Volume 79, Number 386

Theory and Methods Section

355

356 Journal of the American Statistical Association, June 1984

Table 1. Regression Variables Used to Construct a2 With n = 5

Dependent Variable Independent Variable

Y_1 Yo Y1 Y_1 Y2 Yo Y3 Y1 Y4 Y2

Y5 Y3 Y4 Y5

Y5 Y4 Y3 Y5 Y2 Y4

Y1 Y3

Y-1 Y Yo Y_1

Models (1.1) and (2.4) have the property that E( Y,) =

0. A stationary time series with zero mean is seldom en- countered in practice. An alternative to Ho: Ad = 1,

which is reasonable for much real data, is the stationary model in which E( Y,) is nonzero. We therefore consider the regression model

d

Y= ibit + atdYt-d + e,, t = 1, 2, . . ., (2.9) i = I

where

it = 1 if t = i (mod d)

= 0 otherwise,

and {e,} is a sequence of iid (0, cr2) random variables. The regression of Y, on 8 , t,62, ... , d,t, Yt-dfor t = 1, 2,

1 AA

n, produces coefficients 01, 02, O d *, d, ,d. We denote the Studentized regression statistic associated with a,d - 1 by T id.

If we assume that |a(d I <1, a reparameterized version of model (2.9) is

d d

Yt 8it6Li = ad Yt-d - 8itRi + e,, (2.10)

where

Oi - ( - Ud)tiq i = 1, 2, . ,d.

Under model (2.10) the hypothesis Otd = 1 implies that 0i = 0 regardless of the value of ti. Thus, specifying ad

= 1 in the model (2.10) allows pui to assume any value. Under the alternative of I Oad I < 1, however, FL is an identified parameter and should be estimated.

Two estimators of pLj can be considered for the stationary model. The first is that defined by the regression estimators for (2.9), and the second is the seasonal mean [is defined by

ii= (niz ? l)' E -++j j=O

where ni is the greatest integer not exceeding (n + d - i)ld. The estimator ,i can be used to define a symmetric estimator of td analogous to (2.6). We define

n ] I n

=td I (yt2 + yt_d 2 z YtYt-d, (2.12)

where d

Y= Y.,- z Fiit = = 1

If the initial conditions are fixed, the Oi (or [wi) should be estimated using the regression model.

Finally, we consider a model in which the seasonal means ,ui are equal to a constant [L. For the model with constant mean, the ordinary regression estimators are defined by

n

[ ] E~~ Yt-d 0*1 Otd

Yt d E

Yt.-d 2

L*I =[~ ;,= 1K t

x n 1f ld, (2.13) E Yt-d Y.

and the Studentized statistic for Ho: Oxd = 1 is

- nI (E y, sd)j] d* - 1t ),

where n

S*= (n - 2)- E (1K, - - ad* Y,-d) tt=

The symmetric estimator of Ad for the stationary constant-mean model is defined by

n 2 1

n Yt(t + (Y.d*)21} 2 E y.*y.d*,

(2. 14)

where n

S= Y. - 1i . (2.15)

3. LIMITING REPRESENTATION OF REGRESSION STATISTICS

The error in the estimator O 1 is

R1- 1 = {vE yt,,2} 1K Y_e, wea =t1 t=1

when cxl = 1 in m del(.). Dcey (17)baiet

Dickey, Hasza, and Fuller: Seasonal Unit Roots 357

representation for the joint limiting distribution of n n

(n2 I yt_12, n1'f Y-e t = I t = )

and used it to calculate percentiles of the limit distributions of n(LI - 1), Ti, n(&,1 - 1), and T,I by Monte Carlo integration. This approach is now extended to the model with d > 1.

Consider the case d = 2 and the estimator &2. Letting n = 2m, we have

Ot L422 [ Yt-2 ytt2I t t= I ] [t=

-m m ~_ = E Y2J-22+ E y2j132

m m x ? y2i - . (3.1)

i=I i= I

Each summation involves only Y's whose subscripts are even or only Y's whose subscripts are odd. Because

Y2i = eo + + e2 +

and

Y2i_1 = e-1 + el + + e2i_

for all j, we see that summations on even subscripts are independent of summations on odd subscripts. That is,

ot2= [DI + D2]1[N1 + N2],

where in

Nj = E XjtXj,-I, j = 1, 2, (3.2) t= ,

m

Dj = E X1,t1 2, j = 1, 2, (3.3) t= 1

Xj, = Y2, -j + I,

and the vector (NI, D 1) is independent of the vector (N2, D2). The argument extends immediately to the dth order process with estimated means. We have

-d - 1d

Od= Ld D> zi Naj, i= I j=l

where m

N~,j = E Xjt,(Xi, t - I 1-l), j == 1, 2, , d, t= 1

m

D =j = E (Xj, 1 - Xj(- ))2, j = 1, 2, . d, t= I

Xj, = Ydt- j+ 1,

m

XJ(I)~~~~~ = rn

1

and n = md. Therefore, the representations for the limit

distributions given in Theorem 1 follow from the results of Dickey (1976). The asymptotic distributions of the symmetric estimators for ?-d = 1 depend only on the asymptotic distributions of their denominators because, for example,

n

-n-1 , (Yt _ yt _d)2

n(&-d n () -

n2 1: ( yt2 + yt d2)

t = I

n

(n-1 - e 2 t~

n= (3.4) n-2 E ( yt2 + Yd 2)

t= I n

n-l , et 2 >J2 t = 1

and n2 = (yt + Y, + d) has a limiting distribution.

Theorem 1. Let Yt satisfy (1.1), where ad = 1 and {et} is a sequence of iid (0, u 2) random variables. Let &d be defined by (2.1), &' d by regression equation (2.9), & d*

by (2.13), 0d by (2.6), 1t,d by (2.12), and .,, d* by (2.14). Then

L d _I d

n(acd- 1) T 2 BLJ d - T,2

j=l j=l

L d -d 2

Td > Li E (Ti _ 1), 2j=1 j=1

(, -1) L

(Lj _W 2)]

xl(?d [2~> (Tj - w1) TjW

TsJd~~~~~ d [a E L j) 2 (T jW L2

fl(oc d* - 1) ' ~d{d E L1 - (~ wj)}i {j I 2 JE ) =-Ij

f d d 2 I]-

n (.d x L (7>d d 1) Li ( i W ) 9

x ={d 22 - 1)- (d j)( )}

dlO~ -)2 d ( L)

T l(o&d - 1) > - -d2Li (L1 - w2) 1


where

Wj = 2I YkY Zjk, k =1

Tj = 2i I YkZjk, k = 1

k = I

Yk = 2(- )k+ [(2k - 1)7r] 1,

and Zjk are normal independent (0, 1) random variables.

4. ALTERNATIVE REPRESENTATIONS FOR THE LIMIT DISTRIBUTIONS OF THE SYMMETRIC STATISTICS

In Theorem 1 we gave the limiting distribution of all statistics as functions of certain random variables. In this section we present expressions for the limit cumulative distribution functions of the symmetric statistics. These results follow the work of Anderson and Darling (1952), Sargan (1973), Bulmer (1975), and MacNeill (1978). The proofs of Theorems 2-4 are given in Appendix A.

Theorem 2. Let

Xj, = E ejk, k = I

m

Ljm = M2 jX2 t= I

where eJk are iid (0, 1) random variables. Then the limiting distribution of [2nd-2(1 - &Ad)]- 'is the limiting distribution of Ej'= I Ljm, and

d

FL(l) = lim P E Lim C 1} M-30 j= I

= 21(2+d) [F(jd)]l E (- 1)W[F( + 1)]- j= 1

x F(j + Pd) [1 - F{1 1-i(4j + d)}]

for 1 > 0, where ?(x) is the probability that a normal (0, 1) random variable is less than x.

Theorem 3. Let Xj, and Lj. be as defined in Theorem 2. Let

Vjm = Ljm - Wjm2,

m

Wim m- Xt *f= 1

d

Fvd(l) = lim P z Vjm '1l 1>0? m-*oo

j

Then the limiting cumulative distribution function of

[2d-2 n(1 - ,d)]- is FVd(l), and 00

FV2(1) = 4(2,rrl)- , exp{-A1-'(2j + 1)2}, j=O

FV4(1) = 16(21rl3)- z j2 exp{- 21-1I j2}, j=I

FV6(1) = 8(2iTl15)- E (j + 1)(j + 2) j=O

x [(2j + 3)2 - 1]exp{-(21)-'(2j + 3)2}

FV12(l) = 32(15)-'(21r"YI)-i F(j + 3)[r(j - 2)]-1 j=3

x j(1512 - 40j21 + 16j4)exp{-21-1j2}.

Theorem 4. Let the definitions of Theorems 2 and 3 hold. Then the limiting distribution of [2d-2 n(1 - a*pd)] is that of

d

Umd = z Lim - Wim2. J= I

The limiting cumulative distribution function of Umd is

00

Fud(l) = 2 (d-1) i-I 1-i E Aj(4j + d)* j=O

x exp{-(161)-'(4j + d)2}KI[(161)-1(4j + d)2],

where

Aj = Ir4[Fj{(d - 1)}] 2, (_1)k[k!(j k)!]-- k=O

xF{k + l(d - 1)}r{j- k + 1}, j= 0, 1, ...

and K* is the modified Bessel function of the second kind of order one-fourth.

5. PERCENTILES OF SAMPLING DISTRIBUTIONS

Tables of percentiles for the statistics discussed in this article are given in Tables 2-10. For the ordinary regression statistics, the representation in Theorem 1 and the simulation method described by Dickey (1976), and used in Dickey and Fuller (1979,1981) and Hasza (1977), were used to construct limit percentiles. For these limit distributions and for all finite sample sizes, the programs described in Dickey (1981) were used for the Monte Carlo integration. For all combinations of d = 1, 2, 4, 12 and m = 10, 15, 20, 50, 100, 200 and for the limit case, two replications of d-' 60,000 statistics were run. Normal time series were used for all finite sample sizes. The empirical percentiles were smoothed by regression on n - 1. For the symmetric estimators, the regression-smoothed percentiles were forced through the known limit percentiles. The smoothing procedure involved 234 regressions. A lack-of-fit statistic was computed for each regression. Of these, 12 were significant at the 5% level, and four were also significant at 1%.


Table 2. Percentiles for n(aid - 1), the Ordinary Regression Coefficient for the Zero Mean Model

Probability of a Smaller Value

n = md .01 .025 .05 .10 .50 .90 .95 .975 .99

20 -12.20 -9.66 -7.70 -5.62 -.67 1.83 2.43 3.02 3.74 30 -12.67 -9.93 - 7.83 -5.71 -.71 1.72 2.28 2.79 3.43 40 -12.98 -10.11 -7.94 - 5.78 -.73 1.67 2.21 2.69 3.30

d= 2 100 -13.63 -10.50 -8.23 -5.93 -.76 1.58 2.10 2.54 3.07 200 -13.88 -10.65 - 8.34 - 5.99 -.77 1.56 2.06 2.50 3.01 400 -14.01 -10.73 -8.41 - 6.02 -.77 1.54 2.04 2.48 2.97 X0 -14.14 -10.81 -8.47 -6.06 -.78 1.53 2.02 2.46 2.94

40 -13.87 -10.89 -8.67 -6.40 - .61 2.86 3.69 4.41 5.29 60 -14.10 -11.09 -8.86 -6.51 -.65 2.68 3.47 4.13 4.97 80 -14.31 -11.23 -8.94 - 6.54 -.67 2.60 3.38 4.02 4.82

d= 4 200 -14.82 -11.57 -9.08 -6.59 -.70 2.50 3.23 3.85 4.55 400 -15.04 -11.70 -9.12 -6.59 -.71 2.47 3.19 3.80 4.47 800 -15.15 -11.77 -9.14 -6.59 -.71 2.46 3.17 3.78 4.43 x0 -15.27 -11.85 -9.16 -6.59 -.71 2.45 3.15 3.76 4.39

120 -18.10 -14.25 -11.51 -8.76 -.53 5.54 7.02 8.09 9.70 180 -18.04 -14.31 -11.55 -8.74 -.62 5.25 6.66 7.82 9.17 240 -18.02 -14.33 -11.56 -8.73 -.65 5.13 6.52 7.69 8.97

d = 12 600 -18.00 -14.35 -11.57 -8.72 -.69 4.97 6.34 7.49 8.71 1,200 -18.00 -14.35 -11.58 -8.72 -.69 4.93 6.30 7.42 8.65 2,400 -18.00 -14.35 -11.58 -8.72 -.69 4.92 6.28 7.39 8.63

00 - 17.99 -14.35 -11.58 -8.72 -.69 4.90 6.27 7.36 8.61

The leftmost column in each table is headed by n = md, which is the total number of observations. The other column headings are probabilities of obtaining a value smaller than the tabulated value under the model with ad = 1. Percentiles for the Studentized statistics in the symmetric case are not given because the Studentized statistics are simple transformations of the a statistics. The

percentiles for d = 1 are not included in Table 10 because they are identical to those for d = I in Table 9.

The limiting distribution of the symmetric statistic with the mean removed is the same for d = 2 and x2 = 1 as for d = I and ot l = 1. The distribution is also similar for finite n. To see the reason for the similarity, let Y, =

EJ(- I ej and consider the sum of squares for a sample of

Table 3. Percentiles forT'd, the Studentized Statistic for the Ordinary Regression Coefficient of the Zero Mean Model


n = md .01 .025 .05 .10 .50 .90 .95 .975 .99

20 - 2.69 - 2.27 -1.94 -1.57 -.29 1.07 1.48 1.85 2.28 30 - 2.64 - 2.24 -1.93 -1.58 -.31 1.04 1.43 1.79 2.21 40 - 2.62 -2.23 -1.93 -1.58 -.32 1.03 1.42 1.76 2.18

d 2 100 -2.58 -2.22 -1.92 -1.59 -.34 1.01 1.39 1.72 2.12 200 - 2.57 -2.22 -1.92 -1.59 -.35 1.00 1.38 1.71 2.10 400 -2.56 -2.22 -1.92 -1.59 -.35 1.00 1.38 1.70 2.09 X0 - 2.55 - 2.22 -1.92 -1.60 -.35 .99 1.38 1.69 2.08

40 -2.58 - 2.20 -1.87 -1.51 -.20 1.14 1.53 1.86 2.23 60 -2.57 -2.20 -1.89 -1.52 -.21 1.10 1.49 1.82 2.20 80 -2.56 -2.20 -1.89 -1.53 -.22 1.09 1.47 1.80 2.18

d= 4 200 -2.56 -2.20 -1.90 -1.53 -.23 1.07 1.45 1.78 2.17 400 -2.56 -2.20 -1.90 -1.53 -.24 1.07 1.44 1.77 2.16 800 - 2.56 -2.20 -1.90 -1.53 -.24 1.07 1.44 1.77 2.16 X0 - 2.56 - 2.20 -1.90 -1.53 -.24 1.07 1.44 1.77 2.16

120 -2.50 -2.08 -1.77 -1.39 -.10 1.20 1.55 1.87 2.24 180 -2.49 -2.09 -1.77 -1.41 -.12 1.17 1.53 1.85 2.22 240 -2.49 -2.10 -1.77 -1.42 -.13 1.16 1.53 1.85 2.21

d= 12 600 -2.49 -2.10 -1.79 -1.43 -.14 1.15 1.52 1.84 2.20 1,200 - 2.49 - 2.10 -1.79 -1.43 -.14 1.15 1.52 1.84 2.20 2,400 - 2.49 -2.10 -1.80 -1.43 -.14 1.15 1.52 1.84 2.20

00 -2.49 -2.10 -1.80 -1.44 -.14 1.15 1.52 1.84 2.20


Table 4. Percentiles for n(cia,d* - 1), the Ordinary Regression Coefficient for the Single Mean Model


n = md .01 .025 .05 .10 .50 .90 .95 .975 .99

20 -15.57 -13.15 -11.12 -8.97 -2.74 .99 1.75 2.41 3.19 30 -16.82 -13.91 -11.71 - 9.36 - 2.84 .88 1.60 2.21 2.92 40 -17.50 -14.36 -12.04 - 9.56 - 2.89 .84 1.54 2.12 2.79

d = 2 100 -18.81 -15.29 -12.68 -9.96 -2.96 .76 1.44 1.97 2.59 200 -19.27 -15.63 -12.90 -10.10 -2.98 .74 1.41 1.93 2.53 400 -19.50 -15.81 -13.02 -10.18 -2.99 .73 1.39 1.91 2.50 X0 -19.74 -15.99 -13.14 -10.25 - 3.00 .72 1.38 1.89 2.47

40 -17.40 -14.16 -11.66 -9.14 -2.06 2.16 3.10 3.87 4.78 60 -17.85 -14.59 -12.01 -9.27 -2.10 1.98 2.86 3.60r 4.45 80 -18.16 -14.80 -12.17 - 9.35 -2.12 1.91 2.77 3.49 4.31

d = 4 200 -18.90 -15.19 -12.45 -9.52 -2.17 1.81 2.64 3.33 4.11 400 -19.19 -15.32 -12.54 -9.58 -2.19 1.79 2.61 3.29 4.05 800 -19.35 -15.38 -12.58 -9.62 - 2.20 1.78 2.60 3.27 4.02 X0 -19.50 -15.44 -12.62 - 9.65 -2.21 1.78 2.59 3.25 4.00

120 -20.16 -16.35 -13.43 -10.50 -1.67 4.77 6.44 7.63 9.03 180 - 20.28 -16.42 -13.55 -10.47 -1.73 4.52 6.01 7.21 8.65 240 - 20.35 -16.47 -13.59 -10.45 -1.76 4.41 5.85 7.05 8.48

d = 12 600 - 20.47 -16.60 -13.64 -10.44 -1.81 4.27 5.69 6.84 8.19 1,200 - 20.52 -16.65 -13.65 -10.44 -1.82 4.23 5.66 6.80 8.10 2,400 -20.54 -16.68 -13.65 -10.44 -1.83 4.21 5.66 6.78 8.06

oc - 20.56 -16.71 -13.65 -10.44 -1.84 4.20 5.66 6.77 8.02

2n, 2n

k y, )2 (Yt t= 1

2n =n [Yt - _ - ( Y -Yn)]

t = 1

2n n (1Y, - n) - 2n(y - yn)

t= I

= ?E (tj en-j) + z (? en+j)

- (2n)- 'L (E ej - Y, ej ( t=l j= 4 j=)

n-1 r- 2 n r 2 = Es - Eten-i) + en +j r= j=0 r= i= I

-n-I r-I n r 12

- (2n)-' E , (-en_-)+ E en+j . Lr=ij=O r=Ij=_ J

If 2n is replaced by (2n - 1) in the final sum of squares, the resulting quantity has the same distribution as the sum of squares for a sample of 2n - 1 from model (1.1) with d = 2 andt2= 1.

6. HIGHER ORDER MODELS

A popular model for seasonal time series is the multiplicative model

(1 - OtdBd)(l - 01B - - OpBP)Y = e, (6.1)

where B is the backshift operator defined by B( Y,) = Y,- and e, is a sequence of iid (0, CJ2) random variables.

We assume that all roots of

MP - OlmP-' - -p - op = 0 (6.2)

are less than one in absolute value and that Yo, Y_1,. . . . Y-_p-d + are initial values. Hasza and Fuller (1982) considered tests of the hypothesis that Ld = 1 and that one of the roots of (6.2) is one. Model (6.1) defines et as a nonlinear function of (td, 0), where 0' - (0I, 02,

Op). Define this function, evaluated at (cd, 0), by e,(ad, 0). Expanding in Taylor's series, we obtain

e,(&d 0) = edt(d, 0)

-(1 - 01B -02B - O 0PBP) Y.-d(&d - td)

p

-, (Y- - ad Y.-di)(Oi - O) + r,, i= 1

where r, is the Taylor series remainder. This suggests the following estimation procedure (where Y, = - Y- d

corresponds to the initial estimate &Ld = 1):

(1) Regress f, on Y, 1, . . . , Y,_p to obtain an initial estimator of 0 that is consistent for 0 under the null hypothesis that Ad = 1.

(2) Compute e,(1, 0) and regress e,(l, 0) on A 2 A

[(1 - 61B - 02B2 - By - - d,

to obtain estimates of (atd - 1, 0 - 0). The estimator of Oad - 1 may be used to test Ho: td = I

Theorem 5. If ad = 1 in model (6.1), the two-step regression procedure suggested earlier results in an estimator ad and a corresponding Studentized statistic with the same limit distribution as that of the statistic one


Table 5. Percentiles for T,d*, the Studentized Test for the Single Mean Model


n = md .01 .025 .05 .10 .50 .90 .95 .975 .99

20 -3.54 -3.08 - 2.72 - 2.32 -.97 .49 .92 1.31 1.76 30 -3.44 - 3.02 - 2.69 -2.31 -1.01 .46 .88 1.25 1.68 40 -3.40 - 3.00 - 2.68 -2.31 -1.02 .44 .86 1.22 1.65

d = 2 100 -3.31 - 2.95 - 2.65 -2.31 -1.05 .41 .83 1.19 1.61 200 - 3.28 - 2.93 -2.64 -2.31 -1.05 .40 .83 1.19 1.60 400 - 3.27 - 2.93 - 2.64 -2.31 -1.06 .40 .82 1.18 1.59 00 - 3.25 - 2.92 - 2.63 -2.31 -1.06 .40 .82 1.18 1.59

40 - 3.14 - 2.73 -2.38 - 2.00 -.60 .80 1.19 1.54 1.94 60 -3.11 - 2.71 - 2.38 -2.00 -.63 .76 1.15 1.49 1.89 80 -3.09 - 2.71 - 2.38 - 2.01 -.64 .74 1.13 1.47 1.87

d= 4 200 -3.07 -2.70 -2.38 -2.02 -.66 .72 1.11 1.46 1.86 400 - 3.06 - 2.70 -2.38 - 2.02 -.67 .72 1.11 1.45 1.85 800 -3.06 - 2.70 - 2.38 - 2.02 -.67 .72 1.10 1.45 1.85 X0 - 3.05 - 2.70 -2.38 - 2.03 -.67 .72 1.10 1.45 1.85

120 - 2.73 - 2.33 -2.01 -1.65 -.31 1.02 1.38 1.71 2.08 180 - 2.73 - 2.35 - 2.02 -1.66 -.33 .99 1.35 1.68 2.06 240 - 2.73 - 2.36 - 2.02 -1.66 - .34 .98 1.34 1.67 2.05

d = 12 600 - 2.73 - 2.37 - 2.04 -1.66 -.35 .97 1.34 1.66 2.03 1,200 - 2.73 - 2.37 - 2.05 -1.66 -.35 .96 1.33 1.65 2.02 2,400 - 2.74 - 2.37 - 2.06 -1.66 - .36 .96 1.33 1.65 2.02

X0 - 2.74 - 2.37 - 2.06 -1.66 -.36 .96 1.33 1.65 2.01

would obtain by regressing Z, - Z,-d = Zt on Z,d, where Z, = Y, - 0, -- - OP Y,_p. The estimators Oi, obtained by adding the estimates of Oi - Oi to 0i, have the same asymptotic distribution as the coefficients in a regression of Y, on Y,_ , Yt-2, . , ,_p

The proof of Theorem 5 is given in Appendix B. Theo- rem 5 implies that the tabulated limit percentiles for estimators in model (1.1) are applicable in the multiplicative model for large sample sizes.

The extension of Theorem 5 to estimators with seasonal

means or a single mean is immediate. Let

d

Yt =Y- E t iitpi. (6.3)

Replacing Yt by y, in the two-step estimation procedure results in the regression of e,(1, 0) on

(1 - 01B - 02 A

(I - OIB - 02B2 _ 0 ,Bl)yt- d,

kt .

t- .

kt d

Table 6. Percentiles for n(6 ,.d - 1), the Ordinary Regression Coefficient for the Seasonal Means Model


n = md .01 .025 .05 .10 .50 .90 .95 .975 .99

20 -18.98 -16.75 -14.96 -12.78 -6.48 -1.94 - .83 .10 1.16 30 - 20.89 -18.19 -16.01 -13.62 - 6.76 -2.11 -1.01 -.11 .92 40 -22.02 -19.03 -16.64 -14.09 -6.91 -2.18 -1.10 -.21 .79

d = 2 100 - 24.32 - 20.76 -17.95 -15.05 - 7.20 -2.31 -1.25 -.40 .55 200 -25.17 -21.39 -18.44 -15.40 -7.30 - 2.35 -1.30 -.46 .46 400 - 25.62 - 21.72 -18.69 -15.58 - 7.35 - 2.37 -1.33 -.49 .42 00 - 26.07 - 22.06 -18.95 -15.76 -7.41 - 2.39 -1.35 -.52 .38

40 - 28.78 - 25.59 - 23.10 - 20.40 -11.89 - 5.40 - 3.90 - 2.51 -1.05 60 - 30.63 - 27.18 - 24.49 - 21.42 -12.33 - 5.73 -4.15 - 2.81 -1.33 80 -31.79 - 28.12 - 25.27 - 22.00 -12.58 - 5.87 - 4.28 - 2.96 -1.47

d = 4 200 - 34.26 -30.03 - 26.78 - 23.15 -13.06 - 6.09 - 4.51 - 3.20 -1.74 400 -35.20 - 30.73 - 27.32 - 23.56 -13.23 - 6.15 -4.59 - 3.28 -1.83 800 -35.69 -31.10 -27.60 -23.78 -13.32 -6.18 -4.63 -3.32 -1.87 x0 - 36.19 - 31.47 - 27.88 - 24.00 -13.41 - 6.21 - 4.67 - 3.35 -1.92

120 -60.72 -55.63 -51.22 -46.98 -33.65 -22.13 -19.46 -17.09 -14.25 180 -63.40 - 57.83 - 53.57 - 49.14 - 34.95 - 23.06 - 20.00 -17.44 -14.57 240 - 64.90 -59.21 - 54.89 - 50.28 -35.57 - 23.48 - 20.31 -17.70 -14.86

d = 12 600 -67.87 - 62.16 - 57.52 - 52.44 - 36.65 - 24.16 - 20.92 -18.34 -15.57 1,200 - 68.93 - 63.29 - 58.47 - 53.19 - 36.99 - 24.37 - 21.14 -18.60 -15.87 2,400 - 69.48 - 63.87 - 58.95 - 53.57 - 37.16 - 24.47 - 21.26 -18.74 -16.03

xc -70.04 - 64.48 - 59.45 - 53.95 -37.33 - 24.56 - 21.37 -18.88 -16.20


Table 7. Percentiles of Tp,d, the Studentized Statistic for the Seasonal Means Model


n = md .01 .025 .05 .10 .50 .90 .95 .975 .99

20 -4.46 -3.98 -3.60 -3.18 -1.92 -.66 -.29 .03 .42 30 -4.25 -3.84 -3.50 -3.13 -1.95 -.73 -.36 -.04 .34 40 -4.15 -3.77 - 3.45 -3.11 -1.96 -.76 -.40 -.08 .29

d = 2 100 -4.00 -3.66 -3.38 -3.07 -1.99 -.81 -.46 -.15 .21 200 -3.95 -3.63 -3.36 -3.05 -1.99 -.83 -.48 -.18 .18 400 -3.93 -3.62 -3.35 -3.05 -2.00 -.83 -.49 -.19 .17 00 -3.90 - 3.60 - 3.34 -3.04 - 2.00 -.84 -.50 -.20 .15

40 -5.01 -4.57 -4.21 -3.83 -2.55 -1.30 -.94 -.61 -.26 60 -4.85 -4.46 -4.14 -3.79 - 2.58 -1.36 -1.01 -.70 -.35 80 -4.78 -4.41 -4.11 -3.78 -2.60 -1.40 -1.05 -.74 -.39

d = 4 200 -4.67 -4.34 -4.06 -3.75 -2.63 -1.45 -1.11 -.81 -.46 400 -4.64 -4.32 -4.05 -3.74 - 2.64 -1.47 -1.14 -.84 -.48 800 -4.62 -4.31 -4.04 - 3.74 - 2.65 -1.48 -1.15 -.85 -.49 00 -4.61 -4.30 -4.04 -3.73 -2.65 -1.49 -1.16 -.86 -.50

120 - 6.63 - 6.20 -5.86 - 5.49 -4.21 - 2.97 - 2.62 - 2.35 -1.95 180 -6.52 -6.15 -5.84 -5.49 -4.27 -3.06 -2.71 -2.39 -2.01 240 -6.47 -6.13 -5.83 -5.49 -4.30 -3.10 -2.75 -2.43 -2.05

d= 12 600 -6.39 -6.09 -5.82 -5.49 -4.34 -3.17 -2.83 -2.52 -2.18 1,200 -6.37 -6.07 -5.82 -5.49 -4.35 -3.19 -2.86 -2.56 -2.23 2,400 - 6.36 - 6.07 - 5.82 -5.49 -4.36 -3.20 - 2.87 - 2.58 - 2.26

0c -6.35 -6.06 -5.82 -5.49 -4.36 -3.22 - 2.88 - 2.60 - 2.29

Table 8. Percentiles of n(cid - 1), the Symmetric Estimator for the Zero Model

* Probability of a Smaller Value

n= md .01 .025 .05 .10 .50 .90 .95 .975 .99

10 -9.98 -8.31 -6.92 - 5.37 -1.57 -.42 -.33 -.27 -.22 15 -11.18 -9.12 - 7.48 -5.71 -1.62 -.42 -.31 -.25 -.21 20 -11.89 -9.59 - 7.79 -5.90 -1.64 -.42 -.31 -.25 -.20

d = 1 50 -13.38 -10.54 -8.40 -6.27 -1.69 -.42 -.30 -.24 -.19 100 -13.93 -10.89 -8.63 - 6.40 -1.71 -.42 -.30 -.24 -.18 200 -14.22 -11.07 -8.74 -6.46 -1.71 -.42 -.30 -.24 -.18 0c -14.51 -11.26 -8.86 -6.53 -1.72 -.42 -.30 -.23 -.18

20 -12.94 -10.62 -8.71 -6.82 -2.50 -1.01 -.82 -.70 -.60 30 -13.83 -11.11 -9.12 - 7.09 -2.54 -.99 -.79 -.67 -.56 40 -14.29 -11.40 -9.34 - 7.24 - 2.56 -.98 -.78 -.66 -.54

d = 2 100 -15.16 -12.01 -9.75 -7.50 -2.61 -.97 -.77 -.64 -.52 200 -15.46 -12.24 - 9.90 - 7.59 - 2.62 -.97 -.76 -.63 -.51 400 -15.61 -12.36 -9.97 - 7.64 - 2.63 -.97 -.76 -.63 -.51 cc -15.76 -12.48 -10.05 -7.68 -2.64 -.97 -.76 -.63 -.51

40 -16.21 -13.61 -11.45 -9.38 -4.46 - 2.35 -2.01 -1.77 -1.57 60 -16.77 -13.86 -11.73 -9.57 -4.49 -2.31 -1.96 -1.71 -1.49 80 -17.10 -14.06 -11.90 -9.68 -4.51 -2.29 -1.93 -1.68 -1.45

d = 4 200 -17.77 -14.55 -12.23 -9.89 -4.55 - 2.27 -1.90 -1.65 -1.41 400 -18.02 -14.74 -12.35 - 9.97 -4.57 -2.26 -1.90 -1.64 -1.40 800 -18.15 -14.85 -12.41 -10.01 -4.58 - 2.26 -1.89 -1.63 -1.39 cc -18.28 -14.95 -12.48 -10.05 -4.59 -2.26 -1.89 -1.63 -1.39

120 - 26.43 - 23.58 - 21.29 - 18.78 - 12.42 - 8.57 - 7.82 - 7.25 - 6.64 180 - 27.39 - 24.02 - 21.56 - 19.04 - 12.49 - 8.43 - 7.65 - 7.06 - 6.35 240 -27.77 -24.26 -21.72 - 19.18 - 12.52 -8.38 -7.58 -6.98 -6.25

d = 12 600 - 28.29 - 24.70 - 22.07 - 19.44 - 12.55 - 8.32 - 7.48 - 6.85 - 6.13 1200 - 28.42 - 24.86 - 22.20 - 19.53 - 12.55 - 8.31 - 7.46 - 6.82 - 6.12 2400 - 28.47 - 24.94 - 22.27 - 19.57 - 12.55 - 8.31 - 7.45 - 6.80 - 6.11

cc - 28.52 - 25.02 - 22.34 - 19.62 - 12.55 - 8.30 - 7.44 - 6.78 - 6.11

Dlckey, Hasza, and Fuller: Seasonal Unit Roots 363

Table 9. Percentiles of n(a6,d* - 1), the Symmetric Estimator for the Single Mean Model


n = md .01 .025 .05 .10 .50 .90 .95 .975 .99

10 -12.64 -11.28 -9.92 -8.38 -3.84 -1.51 -1.20 -1.01 -.86 15 -14.77 -12.71 -11.01 -9.14 -3.96 -1.47 -1.15 -.96 -.78 20 -15.97 -13.55 -11.62 -9.55 -4.02 -1.46 -1.13 -.93 -.75

d = 1 50 -18.38 -15.30 -12.81 -10.32 -4.13 -1.45 -1.10 -.88 -.70 100 -19.25 -15.95 -13.24 -10.59 -4.17 -1.44 -1.09 -.86 -.69 200 -19.70 -16.28 -13.45 -10.73 -4.19 -1.44 -1.09 -.85 -.68 00 -20.16 -16.62 -13.68 -10.87 -4.21 -1.44 -1.08 -.84 -.67

20 -15.88 -13.48 -11.57 -9.51 -4.02 -1.46 -1.13 -.93 -.75 30 -17.23 -14.45 -12.24 -9.95 -4.08 -1.45 -1.11 -.90 -.72 40 -17.93 -14.97 -12.59 -10.18 -4.11 -1.45 -1.11 -.89 -.71

d = 2 100 -19.25 -15.94 -13.23 -10.59 -4.17 -1.44 -1.09 -.86 -.69 200 -19.70 -16.28 -13.45 -10.73 -4.19 -1.44 -1.09 -.85 -.68 400 -19.93 -16.45 -13.56 -10.80 -4.20 -1.44 -1.09 -.85 -.68 00 -20.16 -16.62 -13.68 -10.87 -4.21 -1.44 -1.08 -.84 -.67

40 -18.97 -16.13 -13.86 -11.45 -5.46 - 2.74 - 2.32 - 2.02 -1.76 60 -19.62 -16.51 -14.08 -11.65 -5.51 - 2.70 - 2.26 -1.97 -1.68 80 -20.03 -16.79 -14.26 -11.77 - 5.53 - 2.69 - 2.24 -1.94 -1.64

d = 4 200 -20.89 -17.47 -14.70 -12.05 -5.59 - 2.67 -2.21 -1.89 -1.60 400 -21.21 -17.74 -14.88 -12.16 -5.61 -2.67 -2.20 -1.87 -1.59 800 -21.38 -17.88 -14.98 -12.21 -5.62 - 2.66 -2.20 -1.86 -1.58 00 -21.56 -18.03 -15.08 -12.27 -5.63 -2.66 -2.20 -1.85 -1.58

120 -28.27 -25.14 - 22.77 - 20.09 -13.15 -9.01 -8.16 -7.51 -6.98 180 -28.91 -25.63 - 22.96 - 20.28 -13.24 -8.87 -8.01 -7.38 -6.70 240 -29.25 -25.92 -23.12 -20.41 -13.28 -8.82 -7.95 -7.30 -6.59

d = 12 600 -29.94 -26.49 -23.51 -20.70 -13.31 -8.76 -7.86 -7.16 -6.45 1,200 -30.18 -26.69 -23.67 -20.81 -13.32 -8.75 - 7.84 -7.11 -6.42 2,400 -30.31 -26.80 -23.76 -20.87 -13.32 -8.75 -7.83 -7.08 -6.40

00 -30.43 -26.91 -23.85 - 20.93 -13.32 -8.75 -7.82 - 7.06 - 6.39

Table 10. Percentiles for n(&a,,d - 1), the Symmetric Estimator for the Seasonal Means Model


n= md .01 .025 .05 .10 .50 .90 .95 .975 .99

20 -19.24 -16.90 -14.97 -12.86 -6.85 -3.46 - 2.92 -2.55 -2.22 30 -21.01 -18.22 -16.04 -13.61 -6.96 -3.39 -2.82 -2.44 -2.07 40 -22.00 -18.96 -16.58 -13.99 -7.02 -3.36 -2.78 -2.38 -2.01

d = 2 100 -23.95 -20.45 -17.59 -14.67 -7.13 -3.32 -2.71 -2.28 - 1.91 200 -24.65 -20.99 -17.93 -14.90 -7.17 -3.31 -2.69 -2.25 -1.89 400 -25.01 -21.26 -18.10 -15.01 -7.19 -3.30 -2.68 -2.23 -1.87 00 -25.37 -21.54 -18.28 -15.13 -7.21 -3.29 -2.68 -2.22 -1.86 40 -28.95 -26.00 -23.23 -20.59 -12.83 -7.88 -6.95 -6.28 -5.58 60 -30.89 -27.35 -24.41 -21.44 -12.90 -7.72 -6.74 -6.03 -5.32 80 -31.85 -28.06 -24.98 -21.84 -12.95 -7.66 -6.65 -5.92 -5.21

d = 4 200 -33.53 -29.41 -25.97 -22.52 -13.08 - 7.57 -6.53 -5.74 -5.03 400 -34.08 -29.88 -26.29 -22.74 -13.13 -7.54 -6.49 -5.68 -4.98 800 -34.35 -30.12 -26.45 - 22.84 -13.16 -7.53 -6.48 -5.65 -4.95 00 -34.63 -30.36 -26.61 -22.95 -13.19 -7.53 -6.47 -5.63 -4.93 120 - 60.83 - 56.03 - 52.50 - 48.69 - 36.64 - 27.56 - 25.69 - 24.02 - 22.39 180 - 63.22 -58.06 - 54.17 - 49.67 - 36.74 - 27.20 - 25.14 - 23.50 - 21 .62 240 - 64.30 - 59.02 - 54.89 - 50.17 - 36.81 - 27.05 - 24.92 - 23.25 - 21.32

d = 12 600 - 66.06 - 60.63 - 56.02 - 51 .07 - 37.00 - 26.87 - 24.61 - 22.79 - 20.90 1 ,200 - 66.59 - 61.14 - 56.35 - 51.37 - 37.08 - 26.82 - 24.54 - 22.64 - 20.81 2,400 - 66.85 - 61.39 - 56.50 - 51.52 - 37.12 - 26.81 - 24.50 -22.57 - 20.76

X0 -67.10 - 61 .64 - 56.65 - 51 .67 - 37.16 - 26.79 - 24.47 - 22.49 - 20.73


Notice that it = y. and that derivatives with respect to Ri in the Taylor series get multiplied by zero, so that no adjustments to ,ui are made in the second step. Using the arguments of Theorem 5, it follows that the first coefficient t1d and its Studentized statistic converge to the limit distributions of the corresponding estimators in model (1.1).

Finally, we extend the results for our symmetric statistics to the multiplicative model. If I ad | < 1 in (6.1), then

(I - aYdFd)(l - 01F - 0pFP)Y

is a white noise series with the same variance as et, where F denotes the forward shift operator F(Yt) = Yt, 1. To compute a symmetric estimator for the model with seasonal means, first compute the deviations from seasonal means defined in (6.3). Then create the column vector

z = (Y-p-d?l, Y-p-d+2, . , Yn Mg

M, . .. 9 Mg Yns . ... 9 Y-p-dJrl).

Here the M's denote a string of p + d missing values. Let the elements of Z be denoted by z,t = - p - d + 1, .. ., 2n + 2(p + d). Now create

AA A A,

e= et(l, 01, 02, O * . 0P) A A A

=(I _-OIB _02B 2 _ .._pBP)Z,

and regress et on

(I - OIB _02B 2 - **-BP)zt-d, Zt, Zt-, 1 , Zt-p

If the first coefficient in this regression is denoted by (&(,d - 1), the percentiles of the limit distribution of n(&,d - 1) are those given in Table 9.

7. POWER STUDY

Using the Monte Carlo method, we computed power curves for the eight test statistics discussed in this paper. Power curves were generated for all combinations of m = 10, 15, 20, 50, 100, and 200, with d = 2, 4, and 12. In all cases, the hypothesis that Otd = 1 was tested at the .05 level against the alternative that ad < 1. The power was evaluated for stationary time series satisfying Y, = at Yt-d + e,, where et are NI(0, 1) random variables for ax = .995, .99, .95, .90, .85, .70, .60, .50, and .30. The empirical power is the fraction of the samples in which

the statistic was in the rejection region. The number of replications was 24,000 for d = 2, 12,000 for d = 4, and 4,000 for d = 12. The power for the eight statistics is given in Table 11 for m = 20 and d = 4.

For m : 100, all statistics have power exceeding .90 for coefficients less than .85. The power increases with d and, of course, with m.

It is interesting that tests based on &d, ci,d*, and &lld* are seriously biased for the stationary alternative with small m and ad close to one. The bias is clear in Table 11. The bias is associated with the properties of stationary time series. For example, if ad = .99, the variance of the stationary time series is 50.25 a2 and the initial value can be far from zero. In small samples with large YO, the variance of the estimator of td iS smaller than the variance of 6d under the null distribution. The T statistics show less bias because these statistics reflect the fact that samples with large Yo have small variance. Note, however, that T,4* is also slightly biased for m = 20 and a4 = .995.

Tests based on statistics constructed with seasonal means removed displayed little bias. For m = 10 and ad

- .995, the null was rejected by the statistic &,4 about 4.8% of the time. For larger m, little or no bias was ob- served. The power of the statistic &4 was similar to that of &t4, with &,4 displaying a slight advantage.

To summarize, if the alternative model is the zero mean or single mean stationary model, the i and TE* statistics are preferred. If the alternative model is the stationary process with seasonal means, (x,, and & j are the most powerful of the statistics studied.

APPENDIX A. PROOFS OF LIMIT RESULTS FOR SYMMETRIC STATISTICS

Proof of Theorem 2. White (1958) showed that

E{exp(2t L 1)} = [cosh(2ti)]-.

Thus

E{exp(-t L1)} = [cosh(2t)VI-1,

andforL = =Li,

E{exp(- t L)} = [cosh(2t)']-Id

= Jfexp(- I t) dFL(l),

Table 11. Empirical Power of Tests Against the Stationary Alternative for 20 Years of Quarterly Data

a4

Statistic .995 .99 .98 .95 .90 .85 .80 .70 .60

&4 .001 .003 .018 .13 .47 .81 .96 1.00 1.00 i4 .090 .114 .170 .35 .66 .88 .97 1.00 1.00 &?L4* .001 .005 .017 .09 .35 .65 .88 1.00 1.00 Tt4* .044 .062 .102 .22 .47 .72 .90 1.00 1.00 a,, 4 .054 .057 .064 .10 .19 .32 .50 .84 .97 TL4 .054 .054 .059 .08 .13 .20 .33 .65 .89 &pA4* .000 .002 .005 .04 .21 .50 .79 .99 1.00 &,>4 .054 .055 .064 .10 .19 .34 .53 .86 .99


where, for notational convenience, we omit the subscript d. Integrating by parts, we have

f exp(-1 t) dFL(l) = t f exp(-1 t)FL(l) dl,

so that

exp(- 1 t)FL(l) dl = t[cosh(2t)id2

By the theory of Laplace transforms, FL(l) is the inverse Laplace transform, Sf -(), given by

FL(l) = I {t [cosh(2t)I] dl2}.

Now

t -'[cosh(2t)I] -Id

= 2Id t- exp[-hd(2t)I] [I + exp(-2(2t)I)] Id

= 2Id -exp[ -d(2t)i] z (-l)'[r(4d) j=0

x r(j + 1)]-' r(j + 2d)exp{-2j(2t)I}.

Therefore,

FL(l) = [r(ld)]' 2id

x , (-l)j[r(j + 1)i-' r(1 + hd)SV-'[f(t)], j=0

where

fit) = t-' exp{-2 -(4j + d)tI}.

From standard tables (Selby 1968, p. 476, No. 83),

[t-' exp{-kt1}] = 2[1 - ?(k(21) 1)],

where '1 is the standard normal cumulative distribution function.

Proof of Theorem 3. We show the general result for V, = Li- Wi2, V = Vi, and d even. Then for every I > O, 1>0,~~~~~~0

Fv(l) = C 2 - Wj(4j + d) j=0

x al exp{-(81)- '(4j + d)2}], d0(mod 4),

d0 k

= C E Wj k[I -l exp{ - (81)- l'(4j + d)2}], j=0

d 2 (mod 4),

where

C = [7rTrF('d)] 23d/4

WI = [F(j + l)]' F1(j + 2 d), (i, k) = ((d - 4)14, (d - 2)14).

Proof. Using results of Anderson and Darling (1952),

E{exp(- t V1)} = [(sinh(2t)l) - (2t)i]i,

and thus

E{exp( - t V)} = [(sinh(2t)i) - (2t)i]i'

x f -exp t} dFv(l)l

By the Laplace transform theory,

Fv(l) = ? -{[t(sinh(2t)I)Wd] 1(2t)/}.

We have

[sinh(2t)l] -dl2 t-1 (20)d/4

= 23d/4 t(d 4)/4 exp[ - d(2t)i2 ] [1 - exp{ - 2(2t)i}] -I

= 23d/4 t(d4)/4 exp[ - d(2t)i2 -] 00

X z [F(j + l)F(' d)] -' F(j + 1 d)exp{- 2j(2t)I}. j=O

Thus

Fv(l) = ,rrjC j W1?I{t(d4)/4exp[-2i(2j + d)ti]}. j=O

The inverse transforms

? -'{exp(- bti)} = [2II1-' b exp{ - b2(41) '},

'- {t-Iexp(-bti)} = (,ml)-Iexp{- b2(41) -},

along with the usual formula for differentiation of Laplace transforms, give the result.

Proof of Theorem 4. Let

d d

U = 1, Li - W12 = (LI - W,2) + a Li, i= I i=2

where d 2 2 is an integer. Letting 2{1} denote the Laplace transform and Fu(I) = P(U ' I), we have

= t '{![exp{(2t)i} + exp{-(2t)I}]}' -d)/2

x {2(2t)l [exp{(2t)I} - exp{ - (2t)I}] `'P = 2(2d+ 1)/4 t- i exp{ - d(2t) 12 -'}

x [1 + exp{-2(2t) }](l-d)2[l - exp{-2(2t)I}]-I

= 2(2d+ l)/4 t- i exp{ - d(2t)i2 -'}

x , (- l)k[k!F((d - 1)/2)] -1 k=O

x F(k + (d - 1)/2)exp{ - 2k(2t)I}

x [j!F(2)] - F(j + )exp{ - 2j(2t)I} j=O

- 2(2d+ 1)/4 2 ajrA-exp{ - ti(4j + d)2 -}, j=O


where

wj = zrVI[l((d - 1)/2] ()k[k!(j - k)!]- k=O

x F(k + (d - 1)12)F(j - k + 1).

The result follows because

Y -'{t-iexp( - t1b)}

= Tr -' bl(21) -exp{ - (81) -' b2}Kj((81)-' b2).

We now establish the relationship between U and n(l - L,,d*). Let Q be a d x d orthonormal matrix with ljth

entry qlj = d-1. Let X, = (Xlt, X2t, , Xdt)'. Let Zt = Q Xt. Notice that d m m

a yaXit2= Y.tr(X ,X,t') j=lt=l t=1

m d m

- z tr(QXtX,'Q') = t 43, t=l j=lt=l

where Zt = (Zlt, Z2t, , Zdt)'. Furthermore, d m m

E E xt= diE Zit, j=lt=l t=1

so that d m d m \2

Xj=21 (dM)-1 (=

i=l t=l j=l t=l d m m 2

'IC' -7 _ Z jt2-(dm)- d a Zit} j=lt=l t=l

Since the covariance matrix of X, is of the form r2 I, so is that of Zt. Thus, the sum of squared deviations of Yt from the single mean

md

Y= (md) Y, Yt t= 1

is md (md )2

yt2 -(md1 EYt t=1 t=1

d m 2

- X2 y d j - E Z1Xjt2 - (md)-

j=1l =l I j=1t= 1

d m m 2

= E zit 2

_-m- I z Zt .

j=lt=l t=l

It follows that the limit distribution of n(I - &,,d*) is the same as that of - d2 {Jd_l Lj _ W1211

APPENDIX B

To prove the results for extension to higher order models, simply note that if

(1 - Bd)(1 - O,B - 02132 - -* -OpBP)Y = e

and if Y, = Y, - Y.-d, then f, is stationary, except for effects of initial conditions, so that

[n -t2 - E( Yn2)] = 0p(n

and

E eeY,1 = 0,(nl),

and by standard theory of stationary time series, ,o - o0 = Op(n -) when Oi is estimated by ordinary least squares from the Y, series.

Now the series A I

_

A 2 A

et= (1 - IB - 02B2 - t - t,

regressed on

(1 - 01B - 02B2- 2

- OPB)Ytd,

yields an estimator of A = Od- 1, denoted by X, and the regression of

et= (1 - 01B - OpBP)kYt

on

(1 - 0IB - p- BP)Y,td

yields the estimator A, where the limit distribution of n A is known.

Now

2 I_-IB - - ...- OpBP)Yt-d]2

- Do - 01B - . PBP- - B))Yt,d]2}

= Op(n-i)

because

n

(Oioi - Oi0j) y2=Op(ni) t= I

Similarly,

n- Ee^t(l - 0IB - 02B2 - - OpBP)Yt -d

-n e l t(I - 0 1 B - '*-OpBP) Yt -d + Op (n-i. A_

Thus n(X - A) = Op(n -), so by Slutzky's theorem, the limit distribution for nA is the same as that of nA. The removal of an overall mean Y at the outset does not affect the estimation of the Oi because (Y, - Y) - ( Ytd - Y) = If - Y,-d = I,, as before. Similarly , etis unchanged but is regressed on (1 - 0AB - OB- PB)(Yt_d - Y). Comparison of this regression coefficient to that in the regression ofeton(1 - OIB - -- - OpBP)(Yt-d - Y) shows, in exactly the same manner as before, that the estimators differ by Op(n -i).

The inclusion of Yt , IYt2, . . . , Yt-p in the regressions allows for Gauss-Newton improvements of 0i but does not affect the limit distribution of n(&d - 1), since E Y,-dY,-j = Op (n).

[Received November 1982. Revised January 1984.1


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testing for unit roots in seasonal time series

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